It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case.
Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.
Let X be a scheme of finite type over a field k. A key goal of algebraic geometry is to compute the Chow groups of X, because they give strong information about all subvarieties of X.
For example, the motivic cohomology groups Hi(X,Z(j)) form a bigraded ring for every scheme X of finite type over a field.
[5] For arbitrary schemes of finite type over a field (not necessarily smooth), there is an analogous spectral sequence from motivic homology to G-theory (the K-theory of coherent sheaves, rather than vector bundles).
Namely, the Beilinson-Lichtenbaum conjecture (Voevodsky's theorem) says that for a smooth scheme X over a field k and m a positive integer invertible in k, the cycle map is an isomorphism for all j ≥ i and is injective for all j ≥ i − 1.
For k a subfield of the complex numbers, a candidate for the abelian category of mixed motives has been defined by Nori.
More precisely, the conjecture predicts the leading coefficient of the L-function at an integer point in terms of regulators and a height pairing on motivic cohomology.
Beilinson and Lichtenbaum made influential conjectures predicting the existence and properties of motivic cohomology.
The definition of higher Chow groups of X is a natural generalization of the definition of Chow groups, involving algebraic cycles on the product of X with affine space which meet a set of hyperplanes (viewed as the faces of a simplex) in the expected dimension.
Voevodsky also defined a motivic cohomology for singular varieties [13] and used it in the proof of the Block-Kato conjecture.